# Projectile: Find range, given two displacement points

• rosemetal17
In summary, the conversation discusses the calculation of the range and initial projection angle of a particle in a horizontal plane, given its initial velocity and two displacement points. The equations used are S = ut + 1/2at^2 and Range = 2u^2sin(alpha)cos(alpha)/g. The attempt at a solution involves solving for u and tan(alpha) using the given equations, but the person is unsure of their algebraic steps. They request help and suspect a problem with the question itself. The expert advises them to show their pair of equations and continue to solve for the angle.
rosemetal17

## Homework Statement

A particle is projected with initial velocity $$u\cos\alpha\vec{i}+u\sin\alpha\vec{j}$$ m/s from a point 0 on a horizontal plane.

If this particle passes through two points whose displacements from 0 are $$3\vec{i}+\vec{j}$$ and $$\vec{i}+3\vec{j}$$

show that the range is $$\frac{13}{4}$$ and that $$\tan\alpha = \frac{13}{3}$$

## Homework Equations

Range

$$\frac{2u^2\sin\alpha\cos\alpha}{g}$$

S, displacement

$$S = ut+\frac{1}{2}at^2$$

## The Attempt at a Solution

So, if the initial velocity is $$u\cos\alpha\vec{i}+u\sin\alpha\vec{j}$$ m/s, then the particle is projected at u m/s at angle $$\alpha$$ to the horizontal.

Then

$$S_x = ut\cos\alpha$$
$$S_y = ut\sin\alpha-\frac{gt^2}{2}$$

and if the displacement points are $$3\vec{i}+\vec{j}$$ and $$\vec{i}+3\vec{j}$$, then:

$$S_x = ut_1\cos\alpha = 3$$
$$S_y = ut_1\sin\alpha-\frac{gt_1^2}{2} = 1$$

and

$$S_x = ut_2\cos\alpha = 1$$
$$S_y = ut_2\sin\alpha-\frac{gt_2^2}{2} = 3$$

I tried solving for the last two pair of equation. For each pair, I eliminated t. So I got two separate equations in total involving $$\alpha$$ and u only.

So here is where I'm stuck.
I tried solving for $$\alpha$$ and u, but I just...can't!? I have no idea what I'm doing wrong, whether if it's algebraic error, or there's something wrong with the formulas. I've been stuck on this for the 5th hour now, and have redone this millions of times.. But still..

I'm beginning to suspect a problem with the question itself! Though I seriously doubt it..

Help'd be muchly appreciated!

You are on the right track. Show your pair of equation for u and tan(alpha) to check. I they are right it is easy to eliminate the term containing u and solving for the angle.

ehild

## 1. What is a projectile?

A projectile is any object that is thrown, shot, or launched into motion and follows a curved path due to the force of gravity acting on it.

## 2. How is the range of a projectile calculated?

The range of a projectile is calculated using the formula R = v02sin(2θ)/g, where R is the range, v0 is the initial velocity, θ is the angle of launch, and g is the acceleration due to gravity.

## 3. What are displacement points?

Displacement points refer to the initial and final positions of a projectile in terms of distance and direction. In the context of finding range, these points would be the initial position where the projectile is launched and the final position where it lands.

## 4. How accurate is the range calculation for a projectile?

The range calculation for a projectile is a simplified model that assumes no air resistance and a constant gravitational force. In real-world scenarios, these factors may affect the accuracy of the calculated range.

## 5. Can the range of a projectile be negative?

Yes, the range of a projectile can be negative if the projectile lands behind the initial position. This can happen if there is a downward slope or the projectile is launched at a high angle with a low initial velocity.

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